Wiegner Eckart theorem, I don't understand it

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In summary, The Wigner-Eckart Theorem states that the matrix elements of quadrupole operators have the same m dependence. The specific equation used for the derivation of the quadrupole effect is Qαβ=∫(3xαxβ-δαβr2)ρdr. This equation is then expressed using the Wigner-Eckart Theorem as <I,m|Qαβ|I,m'>=C<I,m|3/2(IαIβ-IβIα)-δαβI2|I,m'>. The constant C is found using the matrix element for m=m'=I and α=β=z, giving us eQ=<I,I|QzzI,I|>=
  • #1
Jamalll
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Hello,
In derivation of quadrupole effect (which influences NMR spectrum for nuclei with spin ≥ 1/2), there is one step I do not understand, it is Eckart Wiegner theorem, more specific:

just relavant equations:

Qαβ=∫(3xαxβ-δαβr2)ρdr

So here is the W-E:

<I,m|Qαβ|I,m'>=C<I,m|3/2(IαIβ-IβIα)-δαβI2|I,m'>

and we expres constant C witm matrix element for m=m'=I, and α=β=z:

eQ=<I,I|QzzI,I|>=C<I,I|3Iz2-I2|I,I>=<I,I|I(2I-1)|I,I>



questions:
how can we express Qαβ with spin operator?
can someone comment on every step what is being done? I would like to understand this intuitively if possible...Great thanks!
 
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  • #2
Jamalll, What the Wigner-Eckart Theorem says is that the matrix elements of all quadrupole operators have the same m dependence.

Not saying that Q has any simple relationship to I, it's just that 3/2(IαIβ + IβIα) - δαβ is the easiest quadrupole operator to construct and evaluate, and so we use it on the right-hand side to give us something to compare the matrix elements of Q with.
 
  • #3
hey, thanks for reply...


But I still can not make any sense out of it... I know I repeat myself, but
how do you "convert" coordinates in spin operator? What is the significance of "m" and" m' "?

I know it is the state of nucleus with spin "I" and spin projection "m", right?
 
  • #4
Well I am sorry but there is not a single mind which can desolve this?

Baby I am only human!

please...pretty plase
 

1. What is the Wiegner Eckart theorem?

The Wiegner Eckart theorem is a mathematical theorem that relates to the representation theory of a group. It states that for any irreducible representation of a group, there exists a unique vector that is invariant under the action of the group. This vector is known as the Wiegner Eckart vector.

2. How is the Wiegner Eckart theorem applied in science?

The Wiegner Eckart theorem is commonly used in quantum mechanics and spectroscopy to simplify calculations and understand the symmetry properties of molecules. It is also used in nuclear physics and solid-state physics to study the behavior of atomic nuclei and crystals.

3. Can you explain the meaning of the terms "irreducible representation" and "invariant vector" in the Wiegner Eckart theorem?

An irreducible representation is a mathematical concept that describes how a group transforms an object. Invariant vectors are vectors that remain unchanged under the action of a group. In the context of the Wiegner Eckart theorem, the invariant vector is the unique vector that is unaffected by the symmetry operations of the group.

4. What is the significance of the Wiegner Eckart theorem in group theory?

The Wiegner Eckart theorem is an important result in group theory because it provides a way to simplify the analysis of group representations. It also helps in understanding the symmetry properties of physical systems and has applications in various areas of science, including chemistry, physics, and materials science.

5. Are there any limitations or assumptions associated with the Wiegner Eckart theorem?

Yes, the Wiegner Eckart theorem assumes that the group is finite and that the representation is unitary. It also assumes that the group is continuous and that the vector space is finite-dimensional. These assumptions may not hold in all cases, and in such situations, the theorem may not be applicable.

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